CSE 40171: Artificial Intelligence. Informed Search: A* Search

Transcription

1 CSE 40171: Artificial Intelligence Informed Search: A* Search 1

2 Homework #1 has been released. It is due at 11:59PM on 9/10. 2

3 Quick Recap: Search

4 Quick Recap: Search Search problem: States (configurations of the world) Actions and costs Successor function (world dynamics) Start state and goal test Search tree: Vertices: represent plans for reaching states Plans have costs (sum of action costs) Search algorithm: Systematically builds a search tree Chooses an ordering of the fringe (unexplored nodes) Optimal: finds least-cost plans

5 Quick Recap: Search Breadth-First Search Depth-First Search Image credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

6 What was wrong with uninformed search? Did not make use of problem-specific knowledge beyond the definition of the problem itself Was not efficient

7 Image credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188 Informed Search

8 New Aspects of Informed Search The general approach we will consider is best-first search Instance of TREE-SEARCH or GRAPH-SEARCH A vertex is selected for expansion based on an evaluation function, f (v) vertex v with the lowest evaluation is expanded first Most best-first search algorithms include as a component of a heuristic function, h(v) h(v) = estimated cost of the cheapest path from the site at vertex v to a goal state

9 Search Heuristics A heuristic in this context is: A function that estimates how close a state is to a goal Designed for a particular search problem Examples: Manhattan distance, Euclidean distance for path planning Slide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

10 Image Credit: Russel and Norvig h(v)

11 Greedy Best-First Search Image credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

12 Greedy Best-First Search Strategy: expand the vertex that is closest to the goal Assumption: this is likely to lead to a solution quickly Heuristic Function: f (v) = h(v) b Image credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

13 Example with the following heuristic h(v) Image Credit: Russel and Norvig

14 Practical Problems with Greedy Best-First Search A common case: best first takes you straight to the wrong goal b The worst case: like a badly guided depth-first search Sllide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

15 Analysis of Greedy Best-First Search Assume a uniform tree where every state has b successors Completeness: incomplete in a finite state space (just like depth-first search) Optimality: the algorithm is not optimal In our example, we found the path Arad Sibiu Fagaras Bucharest. But this is 32KM longer than the path going from Arad Sibiu Rimnicu Vilcea Pitesti Bucharest. Time complexity: O(b m ), where m is the maximum depth of the search space Space complexity: O(b m )

16 Image credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188 A* Search

17 A* Search A* search is the most widely known form of best-first search g(v): the cost to reach the vertex h(v): the cost to get from the vertex to the goal Vertices are evaluated via: f (v) = g(v) + h(v) i.e., f (v) = estimated cost of the cheapest solution through v

18 Example with the following heuristic g(v): path costs in graph h(v) Image Credit: Russel and Norvig

19 Visual Example of A* Search

20 Conditions for Optimality in TREE-SEARCH: Admissibility Inadmissible (pessimistic) heuristics break optimality by trapping good plans on the fringe Admissible (optimistic) heuristics slow down bad plans but never outweigh true costs Slide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

21 Conditions for Optimality in TREE-SEARCH: Admissibility A heuristic h is admissible (optimistic) if: 0 h(v) h * (v) where h * (v) is the true cost to a nearest goal Example: 15 Coming up with admissible heuristics is most of what is involved in using A* in practice. Slide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

22 Conditions for Optimality in GRAPH-SEARCH: Consistency h(v) is consistent if: For every vertex v and every successor v' of v generated by any action a, the estimated cost of reaching the goal from v is not greater than the step cost of getting v' plus the estimated cost of reaching the goal from v' h(v) c(v, a, v') + h(v') Form of the triangle inequality

23 Optimality of A* TREE-SEARCH Assume: A is an optimal goal vertex B is a suboptimal goal vertex h is admissible Claim: A will exit the fringe before B Slide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

24 Optimality of A* TREE-SEARCH Proof: Assume B is on the fringe Some ancestor n of A is on the fringe, too (possibly A!) Claim: n will be expanded before B 1. f (n) is less or equal to f (A) f (n) = g(n) + h(n) f (n) g(a) g(a) = f (A) Definition of f-cost Admissibility of h h = 0 at a goal Slide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

25 Optimality of A* TREE-SEARCH Proof: Assume B is on the fringe Some ancestor n of A is on the fringe, too (possibly A!) Claim: n will be expanded before B 1. f (n) is less or equal to f (A) 2. f (A) is less than f (B) g(a) < g(b) f (A) < f (B) B is suboptimal h = 0 at a goal Slide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

26 Optimality of A* TREE-SEARCH Proof: Assume B is on the fringe Some ancestor n of A is on the fringe, too (possibly A!) Claim: n will be expanded before B 1. f (n) is less or equal to f (A) 2. f (A) is less than f (B) 3. n expands before B f (n) f (A) < f (B) Slide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

27 Optimality of A* TREE-SEARCH All ancestors of A expand before B A expands before B A* search is optimal Slide credit: Dan Klein and Pieter Abbeel, UC Berkeley CS 188

28 Complexity and Completeness of A* Depth of tree: d Assume a uniform tree where every state has b successors Completeness: A* search is complete Time complexity: O( E ) = O(b d ) Space complexity: O( V ) = O(b d )

29 Applications of A* Search Originally developed as a path planner for Shakey the Robot at Stanford (1968) SRI Shakey with callouts BY-SA 3.0 SRI International

30 Applications of A* Search games-like-age-of-empires-798x350 BY 2.0 Siddhartha Thota Commonly used in games where terrain is mapped to a grid